The least common multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers. To find the LCM of 95 and 90, we can use the following steps:
Step 1: Find All Common Factors
The first step is to find all of the common factors shared by 95 and 90. The common factors of 95 are: 1, 5, 19, 95. The common factors of 90 are: 1, 2, 5, 9, 10, 15, 18, 45, 90.
By comparing these lists, we see that the common factors shared by both 95 and 90 are: 1, 5.
Step 2: Write Out Prime Factorizations
Now we will write out the prime factorizations of each number to help identify the highest common factor:
95 = 5 x 19
90 = 2 x 3 x 3 x 5
The prime factorizations allow us to clearly see that the greatest common factor of 95 and 90 is 5.
Step 3: Multiply Together the Greatest Common Factor and Any Remaining Factors
To find the LCM, we take the greatest common factor (5) and multiply it by any remaining factors from the original numbers.
For 95, the remaining factor is 19.
For 90, the remaining factors are 2, 3, 3.
Therefore:
LCM = Greatest Common Factor x Remaining Factors
= 5 x 19 x 2 x 3 x 3
= 5 x 19 x 18
= 5 x 342
= 1,710
Conclusion
By finding all common factors, writing out the prime factorizations, and multiplying the greatest common factor with the remaining factors, we have determined that the least common multiple of 95 and 90 is 1,710.
| Number | Prime Factorization |
|---|---|
| 95 | 5 x 19 |
| 90 | 2 x 3 x 3 x 5 |
In summary, to find the LCM of two numbers:
- Find all common factors
- Write out the prime factorizations
- Multiply the greatest common factor with any remaining factors
Using this systematic approach allows us to efficiently calculate the LCM of 95 and 90 as 1,710. Understanding how to find the least common multiple is an important math concept with many real-world applications.
Applications of Least Common Multiple
Some examples where knowing how to find the LCM is useful include:
- Determining the lowest common denominator when adding or subtracting fractions
- Finding the smallest container size needed when mixing liquid amounts
- Figuring out the least frequent time interval for repeating events like medication doses
- Calculating minimum runway length needed for takeoff when planes have different requirements
Being able to efficiently find the LCM helps make problem-solving easier in a wide variety of school, work, and everyday situations. Taking a systematic approach breaks the process down into straightforward steps that anyone can follow.
Checking the Work
To double check our work, we can verify that 1,710 is divisible by both 95 and 90:
1,710 / 95 = 18
1,710 / 90 = 19
Since 1,710 is divisible by both original numbers with no remainder, we can confirm it is the least common multiple of 95 and 90.
Practice Finding the LCM
Finding the LCM gets easier with practice. Let’s try a few more examples:
LCM of 12 and 18
- Common factors: 1, 2, 3, 6
- Prime factorizations:
- 12 = 2 x 2 x 3
- 18 = 2 x 3 x 3
- Greatest common factor = 6
- Remaining factors:
- 12: 2
- 18: 3
- LCM = Greatest common factor x Remaining factors
= 6 x 2 x 3
= 36
Therefore, the LCM of 12 and 18 is 36.
LCM of 16 and 24
- Common factors: 1, 2, 4, 8
- Prime factorizations:
- 16 = 2 x 2 x 2 x 2
- 24 = 2 x 2 x 2 x 3
- Greatest common factor = 8
- Remaining factors:
- 16: 2
- 24: 3
- LCM = 8 x 2 x 3 = 48
The LCM of 16 and 24 is 48.
With a bit of practice, finding the least common multiple between two numbers becomes quick and easy. The key steps to remember are finding all common factors, writing out the prime factorizations, and multiplying the greatest common factor by any remaining factors from the original numbers.
Conclusion
In summary, to find the least common multiple of 95 and 90:
- List the factors of each number
- Write out the prime factorizations
- Identify the greatest common factor, 5
- Multiply the greatest common factor by the remaining factors
This results in a least common multiple of 1,710 for the numbers 95 and 90. Understanding how to calculate the LCM is an important math skill that has many real-world applications. With a step-by-step approach and some practice, finding the least common multiple between numbers becomes straightforward. This useful math concept connects to higher level topics like fractions, proportions, and probability.
